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Journal Club April 2010: Negative Poisson's ratio materials

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Metamaterials are artificial materials engineered to provide properties which may not be readily available in nature and are attracting increasing attention. These materials usually gain their properties from structure rather than composition. Although the first metamaterials were photonic (i.e. artificially fabricated, sub-wavelength, periodic structures, designed to interact with optical frequencies) and acoustic (i.e. artificially fabricated materials designed to control, direct, and manipulate sound), the idea may be extended to the mechanical properties of materials defining mechanical metamaterials as artificially fabricated structures designed to achieve unusual mechanical properties. Interesting examples of mechanical metamaterials are provided by negative Poisson’s ratio and negative thermal expansion materials. Here I will focus on negative Poisson’s ratio materials, while inspiring results for negative thermal expansion materials were reported by Sigmund and Torquato (1997) and Steeves et al. (2007).

When materials are compressed (stretched) along a particular axis they are most commonly observed to expand (contract) in directions orthogonal to the applied load (Fig. 1 - left). The property that characterizes this behavior is the Poisson's ratio which is defined as the ratio between the negative transverse and longitudinal strains. The majority of materials are characterized by a positive Poisson’s ratio which is approximately 0.5 for rubber and 0.3 for glass and steel. Materials with a negative Poisson’s ratio will contract (expand) in the transverse direction when compressed (stretched) (Fig. 1 - right) and, although they can exist in principle, demonstration of practical examples is relatively recent.

Figure1: Left: The material elongates and contracts when stretched, leading to a positive Poisson's ratio. Right: The structure unfolds when stretched, causing lateral expansion and a negative Poisson's ratio.

There is significant interest in the development of negative Poisson’s ratio materials because of potential in applications in areas such the design of novel fasteners (Choi and Lakes, 1991), prostheses (Scarpa, 2008), piezocomposites with optimal performance (Sigmund et al, 1998) and foams with superior damping and acoustic properties (Scarpa et al., 2004). Moreover, auxetic materials may lead to the design of stronger composite materials. The primary failure mechanism of composite materials is through reinforcement “pull-out“, a tensile failure caused by the reinforcing fibers getting narrower and pulling away from the matrix. Due to the fact that auxetic materials expand when stretched, however, the load required to cause structural failure will significantly increase.

Discovery and development of materials with negative Poisson’s ratio, also called auxetics, was first reported in the seminal work of Lakes in 1987 (Lakes, 1987). The auxetic behavior was achieved using a novel foam characterized by a reentrant microstructure (Fig. 2) that unfolds when stretched (Fig. 1-right), causing lateral expansion and negative Poisson's ratio. In the case of thermoplastic foams the transformation from the conventional to auxetic form is achieved by triaxial compression followed by heating of the compressed foam to above the softening point.

Negative Poisson’s ration materials occur also in nature. There are a growing number of natural materials that have been discovered to possess one or more negative Poisson's ratios. Baughman et al. (1998) revealed that 69% of the cubic elemental metals and some face-centered cubic (fcc) rare gas solids are auxetic when stretched along the specific [110] off-axis direction. The auxetic effect is correlated with the metal's work function and proposed that auxetic metals could be used as electrodes sandwiching a piezoelectric polymer to give a two-fold increase in piezoelectric device sensitivity.

Moreover, several geometries and mechanisms have been proposed to achieve negative values for the Poisson’s ratio and a variety of man-made auxetic materials and structures have been fabricated and synthesized from the macroscopic down to the molecular, including foams with reentrant structures (Lakes, 1987), hierarchical laminates (Milton, 1992), polymeric and metallic foams (Friis et al, 1988), microporous polymers (Caddock and Evans, 1991), molecular networks (Evans et al, 1991) and many-body systems with isotropic pair interactions (Rechtsman et al, 2008). Negative Poisson’s ratio effects have also been demonstrated at the micron scale using complex materials which were fabricated using soft lithography (Xu et al, 1999) and at the nanoscale with sheets assemblies of carbon nanotubes (Hall et al, 2008). A critical issue related to auxetic material is that their fabrication often requires embedding structures with intricate geometries within a host matrix. However, recently it has been shown that instability induced pattern switches in porous elastomeric structures characterized by an initial simple microstructures may lead to auxetic behavior (Bertoldi e al., 2010).

To conclude I would like to remark the difference between isotropic and anisotropic auxetic materials. From the standpoint of applications Isotropic auxetic materials are more attractive, since negative Poisson’s ratio is achieved for loadings applied along any direction. Energy arguments in the theory of elasticity may be used to show that their Poisson’s ratio for isotropic materials cannot be lower than -1 and larger than ½. Differently, for anisotropic materials the bounds on Poisson's ratio are wider and Poisson's ratios smaller than -1 have been reported.

This journal entry provides a sampling of the research in a field that is active and growing. Discussion on your experiences in this area and on your perceived future challenges is welcomed :)

Firstly, I apologize for the previous comment. I had some problems posting using Chrome & it did not allow me to delete it. I was reading some of the papers that have been linked in the article. It seems a pretty interesting line of research. I had two questions:

1.One of the ways this behavior is explained is by a hinge phenomenon. So as an extensional load is applied, this results in the hinge to open & causes a positive strain in the transverse direction. In that sense there would need to be a theoretical limit to this behavior (i.e. the negative poisson ratio)? If so, wouldn't it be more like two regimes (one where the poisson ratio is negative & then further over a certain load where it might go to a positive poisson ratio regime) or its like a fixed material property?

2. Also how would these materials behave in the case of a cyclic loading (i.e. something like failure by fatigue)?

1) you are right. Energy arguments in the theory of elasticity may be used to show that
the Poisson’s ratio for isotropic materials cannot be lower than -1
and larger than ½.

2) the Poisson's ratio in most of the materials will evolve with deformation. Looking at the structure depicted in fig. 1-left in the Journal entry, we can see that initially when stretched the Poisson's ratio is negative, since it unfolds. However, with increasing deformation the absolute value of the Poisson's ratio will decrease, eventualy becoming positive.

3) About the behavior under cyclic loading: this depends on the material you use to fabricate your negative Poisson's ratio material (i.e rubber, metal etc.)

I find that most papers concern the design of materials with negative Poisson ratio. Thus there are so many types of NPR materials. Are there any more general and more theoretical researches on these materials?

you are totally right. Most of the papers in the field focus either on the design of new materials exhibiting negative Poisson's ratio or on the discovery of natural auxetic materials.

From a theoretical point of view, it is known since a long time that in isotropic materials values of Poisson's ratio larger than one half are thermodynamically inadmissible. Such values would lead to negative strain energy under certain loads.

Katia, for my MilleChili journal and project, I mean for a top performance car, or let's say even for a F1 car, with very large budgets, are these objects any realistic in the near future?

If not what are the limitations to real production? Thanks for answer. Mike

Choi, J. B. and Lakes, R. S., "Design of a fastener
based on negative Poisson's ratio foam", Cellular Polymers, 10,
205-212 (1991).
In this article we make use of the negative Poisson's ratio of recently
developed cellular solids or spongy materials in the design of a
press-fit fastener. Insertion
of the fastener is facilitated by the lateral contraction which negative
Poisson's ratio materials exhibit under compression. Removal of the
fastener
is resisted by the corresponding elastic expansion under tension.Get pdf

Lakes, R. S., "Design considerations for negative Poisson's ratio
materials" ASME Journal of Mechanical Design, 115,
696-700, (1993).
This article presents a study of the implications of negative Poisson's
ratios in the design of load bearing structural elements. Stress
concentration factors are reduced in some situations, and unchanged or
increased in others, when the Poisson's ratio becomes negative. Stress
decay according to Saint Venant's principle can occur more or less
rapidly as the Poisson's ratio decreases. Several design examples are
presented, including a core for a curved sandwich panel and a flexible
impact buffer. Get pdf
Curvature of negative Poisson's ratio honeycomb during bending is convex
in contrast to the saddle shape usually seen for positive Poisson's
ratio, and is shown in the animation at the right and in this video

I have seen the material from the company, but it seems to me the applications are vague, and being 20-30 years that this idea of negative poisson's material is around, I am very worried this may well be a "bluff" !!

Katia, do you have better information about how this is going to work in practise? Which application? I am all in favour of "curiosity - driven" research, do not worry. But the real applications make a huge difference....

In the past years several applications have been identified where the use of negative Poisson's material could lead to a substanital improvment. I am thinking at the biomedical area where -for axample- auxetic materials could be used to design dilators to open up blood vessels during heart surgery. Another potential interesting area relates to the use of auxetic materials in piezoelectric sensors and actuators.

We have seen a number of patent applications from various companies all related to negative posson's ratio materials

but we are still far from the point where negative Poisson's ratio materials will have a real impact in our daily life.

What is the reason for that?

In my opinion the manufacturing process has been a bottleneck in the practical development towards applications. Negative Poisson's ratio materials typically require complex material design and
construction. This motivated my recent work, where instabilities are used to create materials with tunable negative Poisson's ratio. In this wasy we are able to achieveauxetic bahavior starting from a simple microstructure.

Specifically, you assume the problem is manufacturing, the instability-driven manufacturing would be the key revolutionary idea. Hard to beleive, what a coincidence it would be. So your mental process was (i) I have this expertise on instability-driven design, now where to apply it? The best would be neg Poisson's ratio material where this would be beneficial (iii) let's do it!

Or else (i) you were fascinated by neg. poisson's material (ii) were not sure what else to say (iii) thought about why not attempt to use my expertise of instability-by-design materials?

The two make HUGE difference... #2 is much less likely of success. Generalization is an obvious process, generally not revolutionary.

But to tell you precisely, I would need to know more of this "instability"-designed materials. And I would need to be convinced that the problem of auxectic materials is just their manufacturing. I am still worried that their "gain" as sensors or actuator is limited.

In this paper Monte Carlo simulations are used to predicts the nominal elastic constants of thin films comprised of a finite assembly of cubic crystals. Interestingly it is observed that negative values of the effective Poisson’s ratio are realised as a result of the inherent anisotropy of the constituent grains.

I wonder if there's been any work in analyzing defects in metamaterials. If one link breaks, how much do the properties change? Does it have to do with the connectedness of the force-network? If one link breaks, does the instability spread? Furthermore, can one think of imperfections analogous to edge dislocations and write down closed form expressions for resulting stresses as in http://en.wikipedia.org/wiki/Dislocation

I guess it must be a difficult one, but you should be challenged then, instead of giving up already!

I mean is all this fuss about negative Poisson's a mathematical excercise, with curiosity, or is it a real breakthrough which can open new avenues in research, say in cancer, in malaria, in new materials like when the (last) Italian Nobel Prize winner Prof. Giulio Natta invented plastics?

Being at Harvard as a brigth and brilliant italian young promise, I am sure you should be aiming at Nobel prize type of research, not just avenues of research already exhausted. This seems to me quite old stuff, has been postulated to work already 30 years ago if not earlier, companies are already there to try to something (but I suspect the turnover is minimal), so why should we waste more time with it?

Sorry to be direct, I guess you understand what I mean.

If not Natta, lookat Subra Suresh's research across the street from Harvard, and you'll find possibly Nobel prize research. Or, stay at MIT, and look at the other italian brigth guy Francesco Stellacci. Maybe he has elasticity problems which are really worth solving, worth ask him!

Professor Stellacci's research interests are in
nano-science and nano-technology, specifically in the investigation of
the structure-property relationships that exist between nanostructured
molecular assemblies and their surface properties.

His research focuses on the
generation of new understanding on the assembly of molecules in
spatially defined arrangements and their interactions with organic and
bio molecules and with inorganic surfaces. The goal is to apply this
knowledge toward the development and the efficient fabrication of
original nano-size molecular-based materials and devices for a wealth of
applications. In order to build such devices, Stellacci’s group is
developing new materials (organic ligand coated nanoparticles and
nanotubes), and new soft-materials fabrication techniques (based on
molecular recognition and self-assembly). A specific example is the
discovery of novel materials whose outside shell spontaneously assembles
in ways that resemble the structuring of domains on viruses’ capsids.
Another example is the development of a nature-inspired stamping
technique able to transfer DNA patterns from a surface onto another.
This method has been tailored for the efficient production of
inexpensive DNA micro- and nano-arrays. A special emphasis in the group
is placed on the understanding of the nanoscale limitation of present
thermodynamic modeling of surface interactions.

Natta was born in Imperia, Italy. He
earned his degree in chemical engineering from the Politecnico di Milano university in Milan in
1924. In 1927 he passed the exams for becoming a professor there. In
1933 he became a full professor and the director of the Institute of
General Chemistry of Pavia University, where he stayed
until 1935. In that year he was appointed full professor in physical
chemistry at the University of
Rome.

there were a few in science of italian origins but working abroad of course (like it would be your case Katia and Roberto if you manage to be lucky!), but noone like Natta who was working in Italy. Actually Natta is the single one in Chemistry ever!

There are, viceversa, a few in literature, working in Italy. Notice the case of W.D. Phillips of italian origins, and sharing the Nobel prize with Steven Chu present Energy Minister with Obama.... Since it seems extremely difficult to have Carlo Rubbia as Minister of Energy of Berlusconi, despite his innovation in Energy is very interesting, at least we could have been luckier to have Phillips with Obama!

So good luck with Nobel from Negative poisson's ratio. Which area could that be Physics or Chemistry or Medicine?

PHISICS

Carlo Rubbia 1984 "for their decisive contributions to the large project, which led to the discovery of the field particles W and Z, communicators of weak interaction" Rubbia shared the prize with Simon van der Meer.

William D. Phillips 1997 "for development of methods to cool and trap atoms with laser light"
Phillips shared the prize with Steven Chu and Claude Cohen-Tannoudji.

Riccardo Giacconi 2002 "for pioneering contributions to astrophysics, which have led to the discovery of cosmic X-ray sources"Giacconi received half of the prize while Raymond Davis Jr. & Masatoshi Koshiba each received 1/4 of the prize.

Renato Dulbecco 1975 "for their discoveries concerning the interaction between tumour viruses and the genetic material of the cell" Dulbecco shared this award with David Baltimore and Howard Martin Temin.

I work on materials with negative Poisson's ratios (NPR) and negative thermal expansivity (NTE), and it's true that whenever I have what I think is a new idea I go to check Rod Lakes' web page. More often than not I find Rod has done it and published it.... hey ho.

I like your recent work on symmetry breaking. Would you say that breaking of symmetry is neccessary for NPR ? (Am I right in thinking I can draw a honeycomb structure with a negative Poisson's ratio of -1 that does not change its symmetry ?) What do you think that looking at symmetry tells you ? - I'm trying to explore your motivations here.

A significant technical hurdle that remains for NPR materials is their generally low stiffness. Making stiff materials with NPR has eluded everyone so far (to my knowledge). Is there a fundamental reason why this should be so ? (I think there might be). For instance, might you be able to design a fibrous composite which has both high stiffness and NPR via symmetry breaking ? (NPR composites from funny stack sequences have been around for a while but they're all low stiffness).

Nachiket above makes an interesting point about defects in structures that's close to my heart. We have been making NPR mateirals for some time, and some are very homgenous - honeycombs for instance, whereas others were very heterogeneous- foams for instance. We started to wonder how disorder affected Poisson's ratio, and in fact whether it is possible to deliberately use disorder to convert an otherwise+ve Poisson's ratio to -ve. It turns disorder has a strong affect upon Poisson's ratio and it can (probably) be used to generate -ve values 1, 2. Other publications by Neil Gaspar and Curt Koenders also explore this issue from a granluar mechanics standpoint. One thing that has emerged is that adding a small amount of disorder into an otherwise regular structure tends to generate effective structures of far larger scale than the starting base repeating unit. We're looking at some microCT data of deforming NPR foams to see if we can measure this effective structural size.

Concerning NTE - we looked at all the work on NTE and worked out that most if not all of the proposed structures exhibiting NTE could be described as colelctions of hinging or flexing beams. We then wrote down what we think are the base equations for such systems which if you know the geometry and constituent materials of such systems you can use to predict over all CTEs 3. Interestingly CTE is unbounded (a point first made by Rod Lakes) and very large +ve or -ve values are relatively easy to generate.

However many structures proposed for NTE are not stiff or are not weight efficient, something we're trying to investigate. Steeves in his recent JMPS paper addresses the issue of stiffness very well. Certainly the key questions I've been asked by industry on this subject have been about stiffness/strength and weight.

I found the work you pointed out on effect of disorder on the macroscopic properties extremely interesting. It is quite a long time I am thinking at how disorder affect instabilities in periodic structures. In fact, in perfectly periodic structure diffuse modes with a wavelength comparable with the size of the microstructure may take place, while in random microstructures localization occurs first. I find the investigation of the evolution from these two extreme situtations affascinating. In the papers you pointed out, I found both answers and inspiration:)

I am also curious about the stiffness of npr materials. What are typical values of the modulus? In the current state of npr materials, could they even be used in applications such as opening (fairly compliant) blood vessels?

I am not an expert on NPR materials. But is appears to me that they are soft because they are comprised of discrete elements. In the paper I wrote that Katia refers to above, we found NPR in polycrystalline materials with random orientation of anisotropic crystals. I did not pursue these further, but I believe because they are continuous in space they may not suffer from low stiffness as the other types that have been explored.

I think here we are touching another critical issue: stiffness of NPR materials

Typical cellular microstructures are used to achieve NPR, so that - as expected - the stiffness is not that high.

However, as roberto pointed out NPR Poisson's ratio has been found also in polycristalline materials and sheets assemblies of carbon nanotubes (see ref 9 in the review) and this are not typically considered as soft materials.

Another issue could be related to resistance to fatigue. Often NPR is obtained through large rotations inside the microstructure. In the elastomeric samples we tested, we saw that after a while the thin ligements tend to break.

MOLECULAR
DESIGN OF NEW KINDS OF AUXETIC POLYMERS AND …H meiWu Gao-yuanWei - 高分子科学: 英文版, 2004 - cqvip.com
Three
new kinds of molecular networks are designed and predicted to exhibit
negative Poisson
ratios. Molecular mechanics calculations on these networks show that the
magnitude of Poisson
ratios depends on the relative flexibility of beam and arm structures.
Several new kinds of ...Articoli
correlati

I think in certain negative Poisson ratio materials, the stiffness could be high, such as in nacre (ref 18, Phys. Rev. Lett., 245502, 2008). Recently I have a study on a structural composite material inspired by the inner nacreous layer of seashells. We demonstrate this hierarchical mineral/polymer microstructure can be tailored to achieve not only stiffness and strength, but also lateral plastic expansion during tension (a negative “plastic” Poisson’s ratio) providing a volumetric energy dissipation mechanism. However, unlike many other examples of auxetic materials the negative plastic Poisson’s ratio is rarer and has a significant effect in promoting the plastic deformation capability and toughness of the composite.

This is the way forward, keep thinking and speculating of applications! This is the way imechanica is a nice forum in the old sense of Greek and Latin philosophers "forum -- is a latin word " like in the Roman Empire

The Roman Forum, also known by its original Latin designation (Latin: Forum
Romanum, Italian: Foro Romano),
is located between the Palatine
Hill and the Capitoline Hill of the city of Rome, Italy.
Citizens of the ancient city referred to the location as the "Forum
Magnum" or just the "Forum".

Dear Prof Ballarinia - your paper sounds very interesting. Perhaps I'm being stupid but which paper do you refer to ? (I could not see you as an author in any of the papers Prof Bertoldi cites)

Am I right in thinking your argument about 'continuity in space' as being about density? I can't do it now but a survey of NPR materials on a density specific stiffness basis would be revealing. My feeling is they will probably still not compare well.

Prof Bertoldi does have a point, that most NPR materials studied to date have been inherently low modulus. I wonder though if we consider things on a relative basis, do NPRversions have lower moduli values than normal positive PR versions ?

Does anyone know the modulus of the CNT sheets? (My scan of the paper doensn't seem to reveal any).

Fatigue - I would tend to agree that the deformation mechanism leading to NPR (internal rotations) ought to lead to stress concentrations and thus low fatigue life. The only work I know on this is by Fabrizio Scarpa (Bristol) who, if memory serves, found enhanced lifetimes vs conventional versions. Hey ho...

Dr Scarpa is a member of the Aerospace Composites research group. His principal research interests are listed below.

Auxetic structures

Auxetic solids feature a negative Poisson's ratio (NPR) effect, expanding in all directions when pulled in only one. We design and manufacture cellular materials with auxetic characteristics to enhance the structural integrity, vibroacoustic signature and electromagnetic properties of novel concept of sandwich structures. Current programs involve also the design of NPR honeycombs cells with embedded MEMS for structural health monitoring and active electromagnetic compatibility. Cellular structures with NPR capabilities are also used to design novel concepts of morphing airfoils with continuous camber variation.Sponsors: EPSRC, EU FP6, DTI, US Army ARO, QinetiQ plc

Passive and active NPR foams

We have modelled and manufactured samples of negative Poisson’s ratio foams using alternative production procedures from the ones illustrated in literature. Work on the area has focused on static and dynamic performance of mechanical properties, crashworthiness capabilities and absorption acoustic properties. Further studies have been carried out also on auxetic and conventional PU foams doped with magnetorheological fluid – both mechanical, acoustic and dielectric properties showed significant changes when loaded with external magnetic fields.Sponsors: EPSRC, Royal Society, DTI HEFCE, British Vita plc, Xetal ltd

Shape memory alloy honeycombs

We have developed conventional and auxetic honeycombs made of 1 and 2 ways shape memory alloy material. The cellular structures provide changes of stiffness with temperature loading. Large recoverable deformations can be obtained, as well as increased damping capacity under random vibration excitation. The cores can be used in sandwich structures for crashworthiness applications and in satellite-type antennas with deployability capabilities.Sponsors: EPSRC, US Army ARO

Analysis tools

We have developed numerical techniques to improve the vibroacoustic prediction of sandwich and smart structures in middle and high frequency domains, using Spectral Finite Elements or wavelet-based condensation techniques on classical FEM models. Damping characteristics of sandwich structures with viscoelastic material inserts are also simulated with numerical methods improving the initial estimate given by the Modal Strain Energy method applied to classical FEM models. Recent work has focused on homogenisation numerical techniques and the use of metamodelling strategies with Genetic Programming and Artificial Neural Networks to reduce the computational costs for material of microstructure composite design.Sponsors: EPSRC, Rolls Royce plc

Such properties include band gaps and directionality (wave beaming), which can be related to the anisotropic properties of re-entrant lattices, and to the unique deformation mechanism of chiral lattices. A direct relation between NPR and interesting phononic (wave guiding) properties is still missing.

Here is the structure of a 3D auxetic periodic material that uses rotating regular octahedra, from Adrian Rossiter's website, showing how also in this case stress concentration is necessary to achieve NPR. This is somewhat similar to the 2D lattice of connected rotating squares.

All the animations refer to the same structure, the only difference is size and relative motion between bodies. Ignore the attachment/detachment of the vertices during the animation, this is only due to final and initial configurations being identical. Note that the auxetic behaviour is still 2D. Looking carefully, it is possible to spot the global rotational axis. There is no strain along this direction. Considering the fully open configuration, axis Miller indices are [111]. Contractions/expansions take place on its related plane, which is orthogonal to the axis.

Looking even more carefully, only two types of octahedra can be seen, rotating in opposite directions each other.

Although one might be led to think of a global torsional component, this is not present. Each octahedron, excluding those pierced by the rotational axis, moves only radially from its initial position to the outermost position (fully open configuration) and back to tightly pack closer to the axis. This is best shown in the third animation, where the octahedra pierced by the rotational axis are counter-rotating.

Auxetic components of this kind can be manufactured using a Schwarz P surface as a template, and cutting it between two parallel [111] planes.

For a reference on the better-known example of rotating squares, check this article.

Yes. The strain in the direction of the rotational axis (call this z) is zero in any case.

I'm editing my previous post.

Adrian drew my attention on the rotating octahedra being a stack of layers of triangles. I now note that for the auxetic behaviour to be shown, the load should be applied along the principal lattice directions. This is true in any case, 2D (triangles and squares) but also for this 3D sample of rotating octahedra.

Looking through [1 1 1] you can see a hexagon. The three directions joining the vertices of this hexagon are those giving auxetic behaviour (actually their projection on the plane [1 1 1]), namely [2 -1 -1], [-1 2 -1] and [-1 -1 2].

Regarding the constructibility of such a material, here's an example. Consider a cubic sample such as Adrian's oct4. Make the fully-open configuration solid. Remove the spherical joints and thicken the points of contact between octahedra so you can actually build a specimen using an homogeneous material. Something like this below (bottom-right corner sample):

Perhaps it would help to list the groups active in research on NPR and its effects ?

So apart from those people who have already disclosed their interests so far in this thread, I know of the following groups/people -

Anselm Griffin (Georgia Tech, USA)

Andy and Kim Alderson (Bolton Univ, UK)

Fabrizio Scarpa (Bristol, UK)

Krzysztof Wojciechowski (IFM-PAN, Poznan, Poland)

Joseph Grima (Chemistry, Malta)

The order implies nothing, and I accept any blame for ommissions. Please feel free to add to this if you see an ommission.

Would a similar list for materials / structures with negative or unusual CTE be of interest to people ? (The trouble is this would include a lot of chemists and physicists who make/study materials with funny CTE).

I attempted to review this area recently in a paper in Journal of Materials Science (Journal of Materials Science 44 5441-5451 [DOI 10.1007/s10853-009-3692-4]) but I know it's already out of date.

I see the discussion has gone into anisotropic material properties now. It is all interesting, but every now and then, I need to know where we are aiming at. I am still confused about applications. Maybe Katia can help :=)

I'll also point Mike and others at two reviews i) in 2004 by Yang et al in J Mat Sci (vol 30, page 3269), and ii) by Evans and Alderson in Adv Mat in 2000 (vol 12, page 617).

It might be helpfull to divide up applications into those directly associated with the negative Poisson's ratio effect, and those arising indirectly from it.

Direct applications.

Composites with auxetic fibres might show high toughness or be able to use shorter fibres because the auxetic fibres have higher frictional losses during pull out (they lock themselves into their cavities under tension). Such fibres may acts as balancing plies during manufacture because they may effectively reduce thermal expansivity (see recent papers and patents by Alderson). Negative Poisson's ratio in plane may also make draping of prepregs easier because they will naturally form synclastic curvatures (dome shapes) as opposed to anticlastic curvatures (saddle shapes) like positive PR versions.

Some other claims and verifications have been published for fracture toughness in compsoites with various NPRs. I'm not sure there is a large body of evidence for this and this is obviously an interesting avenue of work for someone.

Indentation. since auxetic material underneath a point load (in compression) will tend tomove inwards towards the line of compression rather than away form it as with a PPR material, the resistance to indentation tends to be higher and different with auxetics. This is a nice example of a smart materials response - localised stiffening. I see no reason why such benefits woudl not be seen at high strains rates also, eg for crash protection.

However this response is also in part a result of the increased shear modulus associated with NPR - that is for a given value of Young's moudulus, a NPR below -0.5 will engender a larger shear modulus. As you'll see from the relationship between these elastic constants if nu tends to -1 the shear modulus rises to infinite. Thus Poisson's ratio is a nice way to decouple Young's and shear moduli.

Tunable Deformation. Since the PR controls all deformations in axes other than that applied it offers a large degree of control over how a structure interacts with other structures or fluids. Applications seem possible in aerospace, marine atuomotive and locmotive applications where a particular structural response can be encodedinto astructure via PR especially if it is possible to functionally grade it across a structure.

Indirect applications.

These arise because PR has an influence on other properties or because the microstructure leading to NPR engenders other effects. For example I think both Rod Lakes and Fabrizio Scarpa (and possibly others) have deomonstrated improved acoustic absorption and viscoelastic loss in NPR foams and honeycomb sandwich panels, though in the case of the foams this was probably due to the more tortuous microstructure of the foams than any elastic property per se. We have done some nice work with Fabrizio Scarpa recently on vibration damping in NPR honeycombs which we expect to publish soon.

It has also becaome apparent that there are many advantages in NPR materials as hosts for ElectroMagnetic (EM) active inclussions, usually metallic objects. Control over deformation behaviour in plane in say a honeycomb, allows control over spacing and orientation of EM active elements and thus functionaliies such as absorbtion/relfection, beam steering, wave compression, etc. Perhaps most of this work has so far been done by Fabrizio Scarpa, though I have some ongoing in my laboratory at present. Interstingly, closely related resonance phenomena have been demonstrated for acoustic waves by Massimo Ruzzene (have a look at his webpage for a gorgeous example of beam steering in a chiral NPR honeycomb - http://www.me.gatech.edu/faculty/ruzzene.shtml).

NPR solids such as foams or honeycombs may also have application as separators for particles in fluids. With PPR solids it is possible only to clsoe down pores, but with NPR solids it is possible to open up pores as well. This could have some significant benefits over current filtration systems which usually have to be clsoed down for cleaning or replacement of filters. NPR filters could be cleaned in situ.

There has been some investigation of the benefits of using NPR materials in Piezoceramic composites.

I'm sure I've missed several issues here, so please feel free to point out errors or ommissions.

The rotating triangles work well when there are four
around the tube. The tiling in this case is a kaleidocycle.
This is another mocked up animation, but it would work with
rigid equilateral triangles. It includes the kaleidocycle
motion as well as the rotation of the triangles

I made a animation of Fuller's jitterbug to partner the
rotating octahedron animation. In this case the model
scales equally along the diagonals of the containing
cube (maximum at the icosahedron stage)

However, the rhombic dodecahedron can also be formed by
augmenting the octahedron, and the original "rotating
octahedra" transformation can also be modelled with rhombic
dodecahedra. These pack as before, but are connected
differently